Eigenvalue analysis

Eigenvalues analysis

An automatic linearization of the system is performed at different times of the simulation and all the eigenvalues of the model - time constants and oscillating modes - are listed in a table.

You can use a slider to select the linearization time of interest.

Figure 32: Eigenvalues table

The content of the table can be customized in order to display the properties that are valuable for the model reduction analysis. The configuration button provides the list of values that can be included in the table.

Figure 33: Eigenvalue properties list

Model pertinence

Model pertinence

In order to quickly assess model pertinence, the eigenvalues are color-coded using a double criterion based on the value of their frequency and the value of their damping:

• Red: this color means that the frequency of the eigenvalue is very high compared with a given frequency of interest and this independently of the damping value.

• Yellow: for eigenvalues whose frequencies are not too high compared with the frequency of interest but for which the damping value is lower than a given damping of interest value initially set to 1 %.

• Green: for eigenvalues whose frequency and damping values are respectively low enough compared with the frequency of interest but higher than the damping of interest.

Figure 34: Settings for model pertinence

You should pay a particular attention to red and orange eigenvalues and answer the following questions. Do you know what they represent? Do you really need that level of detail in your model? If not, try to simplify the model to get rid of them.

The default values of the frequency of interest and of the damping of interest are set automatically in order to give sensible pertinent results but you are free to modify these values using the editable fields at the bottom of the eigenvalues list.

The default value of the frequency of interest is equal to the easily observable frequency which is itself based on the print interval (or its equivalent result sampling frequency) set in the Run Parameters of the model. Indeed, it makes sense to base this "interest" on how fast you are sampling/saving the results of your model.

Figure 35: Easily observable frequency in the Run Parameters

The easily observable frequency easily observable frequency (which is equal to the frequency of interest) is defined as a tenth of the result sampling frequency. Indeed, if you want to easily observe a 10 Hz oscillation you are advised to sample the results at a minimum frequency of 100 Hz. The Nyquist-Shannon sampling theorem ensures that when sampling signals at twice the maximum frequency, signals up to 10 Hz can be perfectly reconstructed but it is practical to add a factor 5 to this value as we will illustrate below.

We will take pure sinusoidal signals of 5 Hz, 10 Hz, 25 Hz, 49.9 Hz and 100.1 Hz and sample the results at 100 Hz. Nyquist-Shannon says that signals up to 50 Hz can be perfectly reconstructed but the easily observable frequency is only 10 Hz. Indeed, you can see that frequencies above this value are not easily recognized or could even be invisible (49.9 Hz or 100.1 Hz).

Figure 36: Easily observable frequency in practice

Oscillating modes

Oscillating modes

Oscillating modes are intrinsic properties of the model. With a bit of background in system dynamics they are fairly easy to comprehend. An oscillating mode is a natural frequency or eigenvalue of the system. If the system is excited at this frequency the response will be more or less oscillatory depending on the associated damping.

You should look for higher frequencies which are lightly damped. When they are excited during the simulation, they could slow down the solver. The damping value is also important on its own; a mode with an acceptable frequency and with a very low damping can be more constraining than a higher frequency with a reasonable damping.

The following figure shows a simple second order mechanical model. The input to the system is a step of force of 1 N which is applied to the mass. Since a step has an infinite frequency content it will always excite any frequency in the system.

The natural frequency is set to 0.159 Hz and the damping is varied between 0 and 2 N/(m/s) which is the critical damping for that system. With that critical damping value the mass will respond as fast as it can without oscillating.

Both quantities are computed based on the Real and Imaginary parts of the eigenvalues. The frequencies are computed as follows:

...where is the module of the eigenvalues. The damping ratio is .

Figure 37: Behavior of an oscillating mode depending on damping

Observe how the Oscillating mode becomes a Time constant when the critical damping value is reached.

Time constants

Time constants

Time constants are also fairly easy to understand; they apply to first order systems and are linked to the time delay between the moment when there is a change in the input condition and the moment when the output has fully responded to that change.

Figure 38: Time constant behavior

In this example, the same step (from 0 to 1 at 1 second) is applied as an input to the first order system. The time constant is set to 1 second, and we look at the response of the system. A good approximation is that 95 % of the final value is reached after "3 x time constant" seconds.

Time constants should also be monitored, you can use the linear analysis tool and modal shapes for this purpose. Indeed, very high time constants can also affect solver performance. This becomes clear when observing that a variable will evolve at a very rapid rate and this will necessitate a smaller integration step.

Contribution of state variables

participation factor

In order to have a better understanding of the model, it is very useful to know which components are contributing the most to each eigenfrequency of the system at a linearization time. The top contributing states table lists the top 5 state variables of the model that govern the current selected eigenvalue.

The computation is based on the participation factor that are nondimensional scalars that measure the interaction between the eigenvalues and the state variables.

Figure 39: Top contributing states table

For each particpation factor value the information on the variable title and the associatied submodel is retrieved. Double-click on a line to highlight the component on the sketch.

The content of the table can be customized in order to display the properties that are valuable to identify the contributing states. The configuration button provides the list of attributes that can be included in the table. For instance if one submodel contains state variables with the same title, it is helpful to add the variable names since they are unique per submodel.

Figure 40: Top contributing states properties list

The contributions of the state variables are determined by the modal participation factors that are calculated from the eigenvectors of the linear transformation of the A matrix of the state space model and that measure the interaction between the eigenvalues and the state variables.

You can set a contribution value for which only state variables with a participation factor greater than or equal to this value are displayed in the table. If the option is unchecked, the value is ignored and all state variables are displayed regardless of their contribution.